The celebrated Dutch graphic artist, M. C. Escher, made a unique contribution to the art of pattern-making through his continuous metamorphic designs. His works, Metamorphosis III, or Verbum, show this skill amply. In Metamorphosis III, a long linear scroll, he begins with a simple geometric "day-and-night" (alternating black and white) pattern on the left. As he proceeds to the right, he gradually transforms each "tile" or polygon very slightly. This transformation increases as one moves to the right and eventually the original tiles completely change to another set of tiles. One pattern changes to another in the process. In Metamorphosis III he does this continually and goes from one pattern change to another in the same illustration.
Metamorphic tiling patterns provide useful and visually interesting applications as architectural patterns in buildings, as floor and wall tiles, as ceiling lattices or window screens, as partitions, textile patterns, layout of buildings or in landscape designs. The tiling patterns could be used in various crafts, art works, brick designs, or as toys and puzzles.
Prior art include's Escher's metamorphic tiling patterns which are well known from his graphic prints and publications on his work. Prior art, like Escher's, is restricted to linear transformations, i.e. transformations along one direction as in Escher's Metamorphosis III, transformational patterns on a square, i.e. transformations along two simultaneous directions, and transformational patterns on a regular hexagon, i.e. transformations along three directions as in Escher's Verbum. The use of higher dimensions for deriving transformational tiling patterns is not known in prior art. The present invention shows a generalization of metamorphic tiling patterns by projection from n-dimensions into 2- or 3-dimensions. This is not trivial. The present invention uses 2-dimensional projection of an n-dimensional cube as an underlying or "hidden" network, hereafter termed "network", for deriving continuous pattern transformations. The tilings derived can be termed "Hyper-Escher" patterns.
More specifically, zonogons (in 2-dimensions) and zonohedra (in 3-dimensions), which are embedded in the n-cube and are like its "shadows" are used as networks instead of the entire n-dimensional cube. This is to avoid over lapping tiling patterns which will result if the entire n-cube were used. In the 2-dimensional case this leads to zonogons, or 2n-sided polygons having their opposite edges parallel to one another, which are divided into different rhombii or paralellograms. When divided thus, the zonogon is in fact a 2-dimensional view of a zonohedron, a polyhedron with n(n-1) faces in parallel pairs. This zonohedron is used as network to generate Escher-like metamorphic designs. Since n can be any number, such patterns are an infinite class. In the 3-dimensional case, the rhombic or paralellogram faces of a zonohedron, are used as a starting point.
The tiling patterns could be suitably colored. The color scheme could itself reflect the idea of metamorphosis and the tiles could be graded in color. This means n tranformations would require n different colors in binary combinations. Thus, as the shape of the tile changes, so does its color.
One example of the derivation of metamorphic tiling patterns using this method is described in detail. This example shows a tiling based on 4 transformations on a single edge of a tile. In addition, the tiles shown in this particular example are all 4-sided. The array of these 4-sided polygons uses a "base" square grid (shown later in FIG. 10 by a graph, and in FIGS. 11 and 13 by an array of black dots). Each "base polygon" of this grid is a square. This "base grid" is also hidden and is superimposed on the zonohedron network. Further, in the example shown, the zonohedron network has a true 4-fold symmetry which happens to match with the symmetry of the base grid.
It will be clear that other matamorphic tilings can be derived in this manner. The base polygons need not be squares, and any rectangle, rhombus or a parallelogram could be used. In addition, the base grid need not be a square grid and could be based on the arrays of different base polygons. The edges could use other types of transformations and could be curved in various ways. The tile could be made 3-dimensional in various ways. The zonohedron network could use other paralellograms or rhombii with different angles, and its dimension could be greater than 4.